A characterization of superreflexivity through embeddings of lamplighter groups

TL;DR

This paper shows that finite lamplighter groups can be embedded with bounded distortion into any non-superreflexive Banach space, establishing them as test-spaces for superreflexivity.

Contribution

It introduces a new characterization of superreflexivity using embeddings of lamplighter groups, inspired by their Cayley graph structure as horocyclic products of trees.

Findings

Finite lamplighter groups embed with bounded distortion into non-superreflexive Banach spaces.

Lamplighter groups serve as test-spaces for superreflexivity.

The proof uses a novel covering approach inspired by horocyclic products of trees.

Abstract

We prove that finite lamplighter groups $\{\mathbb{Z}_2\wr\mathbb{Z}_n\}_{n\ge 2}$ with a standard set of generators embed with uniformly bounded distortions into any non-superreflexive Banach space, and therefore form a set of test-spaces for superreflexivity. Our proof is inspired by the well known identification of Cayley graphs of infinite lamplighter groups with the horocyclic product of trees. We cover $\mathbb{Z}_2\wr\mathbb{Z}_n$ by three sets with a structure similar to a horocyclic product of trees, which enables us to construct well-controlled embeddings.